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A non-local bistable reaction-diffusion equation with a gap

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  • Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t-∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

    Mathematics Subject Classification: 35B08, 35B50, 35K57, 35R09.


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  • Figure 1.  Results from numerical simulations for the gap region problem with probability distribution satisfying $(H1)$ with $\sigma =3$ and $\sigma=5$. Figure (1a) displays a gap region of size 10. In Figure (1b) the length of the gap region is 15. Note that the gap has been shifted for the numerical computations.

    Figure 2.  Illustration of the change in the right-most value of the support of the subsolutions. The $x-axis$ provides the center of the support of the subsolutions. Figure 2a illustrates a case when $L=5.$ Here, one only observes the expected discontinuities, when $x=0$ (corresponding to $(i)$ in the figure) and when $x=L$ enter the domain (corresponding to $(ii)$ in the Figure). Figure 2b illustrates the case when $L=12$. On the contrary to the previous case, we see that there is an additional discontinuity (corresponding to $(iii)$)

    Figure 3.  Numerical solutions to the Cauchy problem (2.10) with a step function initial condition at time $t=200$. Figure 3a illustrates the case when the gap is too small to prevent propagation (L = 5). Figure 3a illustrates the case when the gap is sufficiently large to obstruct propagation (L = 12).

    Figure 4.  Family of subsolutions as the domain is shifted from left to right. Here, $L=12$ and the discontinuity can be seen clearly in sequence of subfigures 8-10.

    Figure 5.  Family of subsolutions as the domain is shifted from left to right. Here, L = 12 and the discontinuity can be seen clearly in sequence of subfigures 8-10.

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